- FindStat

Identifier

Mp00018: Binary trees —left border symmetry⟶ Binary trees
St000398: Binary trees ⟶ ℤ

Values

[.,.] => [.,.] => 0

[.,[.,.]] => [.,[.,.]] => 1

[[.,.],.] => [[.,.],.] => 1

[.,[.,[.,.]]] => [.,[.,[.,.]]] => 3

[.,[[.,.],.]] => [.,[[.,.],.]] => 3

[[.,.],[.,.]] => [[.,[.,.]],.] => 3

[[.,[.,.]],.] => [[.,.],[.,.]] => 2

[[[.,.],.],.] => [[[.,.],.],.] => 3

[.,[.,[.,[.,.]]]] => [.,[.,[.,[.,.]]]] => 6

[.,[.,[[.,.],.]]] => [.,[.,[[.,.],.]]] => 6

[.,[[.,.],[.,.]]] => [.,[[.,[.,.]],.]] => 6

[.,[[.,[.,.]],.]] => [.,[[.,.],[.,.]]] => 5

[.,[[[.,.],.],.]] => [.,[[[.,.],.],.]] => 6

[[.,.],[.,[.,.]]] => [[.,[.,[.,.]]],.] => 6

[[.,.],[[.,.],.]] => [[.,[[.,.],.]],.] => 6

[[.,[.,.]],[.,.]] => [[.,[.,.]],[.,.]] => 4

[[[.,.],.],[.,.]] => [[[.,[.,.]],.],.] => 6

[[.,[.,[.,.]]],.] => [[.,.],[.,[.,.]]] => 4

[[.,[[.,.],.]],.] => [[.,.],[[.,.],.]] => 4

[[[.,.],[.,.]],.] => [[[.,.],[.,.]],.] => 5

[[[.,[.,.]],.],.] => [[[.,.],.],[.,.]] => 4

[[[[.,.],.],.],.] => [[[[.,.],.],.],.] => 6

[.,[.,[.,[.,[.,.]]]]] => [.,[.,[.,[.,[.,.]]]]] => 10

[.,[.,[.,[[.,.],.]]]] => [.,[.,[.,[[.,.],.]]]] => 10

[.,[.,[[.,.],[.,.]]]] => [.,[.,[[.,[.,.]],.]]] => 10

[.,[.,[[.,[.,.]],.]]] => [.,[.,[[.,.],[.,.]]]] => 9

[.,[.,[[[.,.],.],.]]] => [.,[.,[[[.,.],.],.]]] => 10

[.,[[.,.],[.,[.,.]]]] => [.,[[.,[.,[.,.]]],.]] => 10

[.,[[.,.],[[.,.],.]]] => [.,[[.,[[.,.],.]],.]] => 10

[.,[[.,[.,.]],[.,.]]] => [.,[[.,[.,.]],[.,.]]] => 8

[.,[[[.,.],.],[.,.]]] => [.,[[[.,[.,.]],.],.]] => 10

[.,[[.,[.,[.,.]]],.]] => [.,[[.,.],[.,[.,.]]]] => 8

[.,[[.,[[.,.],.]],.]] => [.,[[.,.],[[.,.],.]]] => 8

[.,[[[.,.],[.,.]],.]] => [.,[[[.,.],[.,.]],.]] => 9

[.,[[[.,[.,.]],.],.]] => [.,[[[.,.],.],[.,.]]] => 8

[.,[[[[.,.],.],.],.]] => [.,[[[[.,.],.],.],.]] => 10

[[.,.],[.,[.,[.,.]]]] => [[.,[.,[.,[.,.]]]],.] => 10

[[.,.],[.,[[.,.],.]]] => [[.,[.,[[.,.],.]]],.] => 10

[[.,.],[[.,.],[.,.]]] => [[.,[[.,[.,.]],.]],.] => 10

[[.,.],[[.,[.,.]],.]] => [[.,[[.,.],[.,.]]],.] => 9

[[.,.],[[[.,.],.],.]] => [[.,[[[.,.],.],.]],.] => 10

[[.,[.,.]],[.,[.,.]]] => [[.,[.,[.,.]]],[.,.]] => 7

[[.,[.,.]],[[.,.],.]] => [[.,[[.,.],.]],[.,.]] => 7

[[[.,.],.],[.,[.,.]]] => [[[.,[.,[.,.]]],.],.] => 10

[[[.,.],.],[[.,.],.]] => [[[.,[[.,.],.]],.],.] => 10

[[.,[.,[.,.]]],[.,.]] => [[.,[.,.]],[.,[.,.]]] => 6

[[.,[[.,.],.]],[.,.]] => [[.,[.,.]],[[.,.],.]] => 6

[[[.,.],[.,.]],[.,.]] => [[[.,[.,.]],[.,.]],.] => 8

[[[.,[.,.]],.],[.,.]] => [[[.,[.,.]],.],[.,.]] => 7

[[[[.,.],.],.],[.,.]] => [[[[.,[.,.]],.],.],.] => 10

[[.,[.,[.,[.,.]]]],.] => [[.,.],[.,[.,[.,.]]]] => 7

[[.,[.,[[.,.],.]]],.] => [[.,.],[.,[[.,.],.]]] => 7

[[.,[[.,.],[.,.]]],.] => [[.,.],[[.,[.,.]],.]] => 7

[[.,[[.,[.,.]],.]],.] => [[.,.],[[.,.],[.,.]]] => 6

[[.,[[[.,.],.],.]],.] => [[.,.],[[[.,.],.],.]] => 7

[[[.,.],[.,[.,.]]],.] => [[[.,.],[.,[.,.]]],.] => 8

[[[.,.],[[.,.],.]],.] => [[[.,.],[[.,.],.]],.] => 8

[[[.,[.,.]],[.,.]],.] => [[[.,.],[.,.]],[.,.]] => 6

[[[[.,.],.],[.,.]],.] => [[[[.,.],[.,.]],.],.] => 9

[[[.,[.,[.,.]]],.],.] => [[[.,.],.],[.,[.,.]]] => 6

[[[.,[[.,.],.]],.],.] => [[[.,.],.],[[.,.],.]] => 6

[[[[.,.],[.,.]],.],.] => [[[[.,.],.],[.,.]],.] => 8

[[[[.,[.,.]],.],.],.] => [[[[.,.],.],.],[.,.]] => 7

[[[[[.,.],.],.],.],.] => [[[[[.,.],.],.],.],.] => 10

[.,[.,[.,[.,[.,[.,.]]]]]] => [.,[.,[.,[.,[.,[.,.]]]]]] => 15

[.,[.,[.,[.,[[.,.],.]]]]] => [.,[.,[.,[.,[[.,.],.]]]]] => 15

[.,[.,[.,[[.,.],[.,.]]]]] => [.,[.,[.,[[.,[.,.]],.]]]] => 15

[.,[.,[.,[[.,[.,.]],.]]]] => [.,[.,[.,[[.,.],[.,.]]]]] => 14

[.,[.,[.,[[[.,.],.],.]]]] => [.,[.,[.,[[[.,.],.],.]]]] => 15

[.,[.,[[.,.],[.,[.,.]]]]] => [.,[.,[[.,[.,[.,.]]],.]]] => 15

[.,[.,[[.,.],[[.,.],.]]]] => [.,[.,[[.,[[.,.],.]],.]]] => 15

[.,[.,[[.,[.,.]],[.,.]]]] => [.,[.,[[.,[.,.]],[.,.]]]] => 13

[.,[.,[[[.,.],.],[.,.]]]] => [.,[.,[[[.,[.,.]],.],.]]] => 15

[.,[.,[[.,[.,[.,.]]],.]]] => [.,[.,[[.,.],[.,[.,.]]]]] => 13

[.,[.,[[.,[[.,.],.]],.]]] => [.,[.,[[.,.],[[.,.],.]]]] => 13

[.,[.,[[[.,.],[.,.]],.]]] => [.,[.,[[[.,.],[.,.]],.]]] => 14

[.,[.,[[[.,[.,.]],.],.]]] => [.,[.,[[[.,.],.],[.,.]]]] => 13

[.,[.,[[[[.,.],.],.],.]]] => [.,[.,[[[[.,.],.],.],.]]] => 15

[.,[[.,.],[.,[.,[.,.]]]]] => [.,[[.,[.,[.,[.,.]]]],.]] => 15

[.,[[.,.],[.,[[.,.],.]]]] => [.,[[.,[.,[[.,.],.]]],.]] => 15

[.,[[.,.],[[.,.],[.,.]]]] => [.,[[.,[[.,[.,.]],.]],.]] => 15

[.,[[.,.],[[.,[.,.]],.]]] => [.,[[.,[[.,.],[.,.]]],.]] => 14

[.,[[.,.],[[[.,.],.],.]]] => [.,[[.,[[[.,.],.],.]],.]] => 15

[.,[[.,[.,.]],[.,[.,.]]]] => [.,[[.,[.,[.,.]]],[.,.]]] => 12

[.,[[.,[.,.]],[[.,.],.]]] => [.,[[.,[[.,.],.]],[.,.]]] => 12

[.,[[[.,.],.],[.,[.,.]]]] => [.,[[[.,[.,[.,.]]],.],.]] => 15

[.,[[[.,.],.],[[.,.],.]]] => [.,[[[.,[[.,.],.]],.],.]] => 15

[.,[[.,[.,[.,.]]],[.,.]]] => [.,[[.,[.,.]],[.,[.,.]]]] => 11

[.,[[.,[[.,.],.]],[.,.]]] => [.,[[.,[.,.]],[[.,.],.]]] => 11

[.,[[[.,.],[.,.]],[.,.]]] => [.,[[[.,[.,.]],[.,.]],.]] => 13

[.,[[[.,[.,.]],.],[.,.]]] => [.,[[[.,[.,.]],.],[.,.]]] => 12

[.,[[[[.,.],.],.],[.,.]]] => [.,[[[[.,[.,.]],.],.],.]] => 15

[.,[[.,[.,[.,[.,.]]]],.]] => [.,[[.,.],[.,[.,[.,.]]]]] => 12

[.,[[.,[.,[[.,.],.]]],.]] => [.,[[.,.],[.,[[.,.],.]]]] => 12

[.,[[.,[[.,.],[.,.]]],.]] => [.,[[.,.],[[.,[.,.]],.]]] => 12

[.,[[.,[[.,[.,.]],.]],.]] => [.,[[.,.],[[.,.],[.,.]]]] => 11

[.,[[.,[[[.,.],.],.]],.]] => [.,[[.,.],[[[.,.],.],.]]] => 12

[.,[[[.,.],[.,[.,.]]],.]] => [.,[[[.,.],[.,[.,.]]],.]] => 13

[.,[[[.,.],[[.,.],.]],.]] => [.,[[[.,.],[[.,.],.]],.]] => 13

[.,[[[.,[.,.]],[.,.]],.]] => [.,[[[.,.],[.,.]],[.,.]]] => 11

[.,[[[[.,.],.],[.,.]],.]] => [.,[[[[.,.],[.,.]],.],.]] => 14

>>> Load all 1200 entries. <<<

search for individual values

searching the database for the individual values of this statistic

/ search for generating function

searching the database for statistics with the same generating function

click to show known generating functions

Description

The sum of the depths of the vertices (or total internal path length) of a binary tree.
The depth of a vertex is the number of edges to the tree's root, see Section 2.3.4.5 of [1] and [3].
This statistic is the very first entry of the OEIS, see [2]. Observe that there the term height is used instead.

Map

left border symmetry

Description

Return the tree where a symmetry has been applied recursively on all left borders. If a tree is made of three trees $T_1, T_2, T_3$ on its left border, it becomes $T_3, T_2, T_1$ where same symmetry has been applied to $T_1, T_2, T_3$.